PART 2 Calculating Discount Rates
118
PRIOR RESEARCH
Historically, small companies have shown higher rates of return when
compared to large ones, as evidenced by data for the New York Stock
Exchange (NYSE) over the past 73 years of its existence (Ibbotson Asso-
ciates 1999). The relationship between ´¬ürm size and rate of return was
´¬ürst published by Rolf Banz in 1981 and is now universally recognized.
Accordingly, company size has been included as a variable in several
models used to determine stock market returns.
Jacobs and Levy (1988) examined small ´¬ürm size as one of 25 vari-
ables associated with anomalous rates of return on stocks. They found
that small size was statistically signi´¬ücant both in single-variable and
multivariate form, although size effects appear to change over time, i.e.,
they are nonstationary. They found that the natural logarithm (log) of
market capitalization was negatively related to the rate of return.
Fama and French (1993) found they could explain historical market
returns well with a three-factor multiple regression model using ´¬ürm size,
the ratio of book equity to market equity (BE/ME), and the overall market
factor Rm Rf , i.e., the equity premium. The latter factor explained overall
returns to stocks across the board, but it did not explain differences from
one stock to another, or more precisely, from one portfolio to another.2
The entire variation in portfolio returns was explained by the ´¬ürst
two factors. Fama and French found BE/ME to be the more signi´¬ücant
factor in explaining the cross-sectional difference in returns, with ´¬ürm size
next; however, they consider both factors as proxies for risk. Furthermore,
they state, ÔÇ˜ÔÇ˜Without a theory that speci´¬ües the exact form of the state
variables or common factors in returns, the choice of any particular ver-
sion of the factors is somewhat arbitrary. Thus detailed stories for the
slopes and average premiums associated with particular versions of the
factors are suggestive, but never de´¬ünitive.ÔÇ™ÔÇ™
Abrams (1994) showed strong statistical evidence that returns are
linearly related to the natural logarithm of the value of the ´¬ürm, as mea-
sured by market capitalization. He used this relationship to determine the
appropriate discount rate for privately held ´¬ürms. In a follow-up article,
Abrams (1997) further simpli´¬üed the calculations by relating the natural
log of size to total return without splitting the result into the risk-free
rate plus the equity premium.
Grabowski and King (1995) also described the logarithmic relation-
ship between ´¬ürm size and market return. They later (Grabowski and
King 1996) demonstrated that a similar, but weaker, logarithmic relation-
ship exists for other measures of ´¬ürm size, including the book value of
common equity, ´¬üve-year average net income, market value of invested
capital, ´¬üve-year average EBITDA, sales, and number of employees. Their
latest research (Grabowski and King 1999) demonstrates a negative log-
arithmic relationship between returns and operating margin and a posi-

2. The regression coef´¬ücient is essentially beta controlled for size and BE/ME. After controlling for
the other two systematic variables, this beta is very close to 1 and explains only the market
premium overall. It does not explain any differentials in premiums across ´¬ürms or
portfolios, as the variation was insigni´¬ücant.

CHAPTER 4 Discount Rates as a Function of Log Size 119
tive logarithmic relationship between returns and the coef´¬ücient of vari-
ation of operating margin and accounting return on equity.
The discovery that return (the discount rate) has a negative linear
relationship to the natural logarithm of the value of the ´¬ürm means that
the value of the ´¬ürm decays exponentially with increasing rates of return.
We will also show that ´¬ürm value decays exponentially with the standard
deviation of returns.

TABLE 4-1: ANALYSIS OF HISTORICAL STOCK RETURNS
Columns AÔÇ“F in Table 4-1 contain the input data from the Stocks, Bonds,
Bills and In´¬‚ation 1999 Yearbook (Ibbotson Associates 1999) for all of the
regression analyses as well as the regression results. We use the 73-year
average arithmetic returns in both regressions, from 1926 to 1998. For
simplicity, we have collapsed 730 data points (73 years 10 deciles) into
73 data points by using averages. Thus, the regressions are cross-sectional
rather than time series. Column A lists the entire NYSE divided into dif-
ferent groups (known as deciles) based on market capitalization as a
proxy for size, with the largest ´¬ürms in decile #1 and the smallest in decile
10.3 Columns B through F contain market data for each decile which is
described below.
Note that the 73-year average market return in Column B rises with
each decile. The standard deviation of returns (Column C) also rises with
each decile. Column D shows the 1998 market capitalization of each dec-
ile, with decile #1 containing 189 ´¬ürms (Column F) with a market capi-
talization of $5.986 trillion (D8). Market capitalization is the price per
share times the number of shares. We use it as a proxy for the fair market
value (FMV).
Dividing Column D (FMV) by Column F (the number of ´¬ürms in the
decile), we obtain Column G, the average capitalization, or the average
fair market value of the ´¬ürms in each decile. For example, the average
company in decile #1 has an FMV of $31.670 billion (G8, rounded), while
the average ´¬ürm in decile #10 has an FMV of $56.654 million (G17,
rounded).
Column H shows the percentage difference between each successive
decile. For example, the average ´¬ürm size in decile #9 ($146.3 million;
G16) is 158.2% (H16) larger than the average ´¬ürm size in decile #10 ($56.7
million; G17). The average ´¬ürm size in decile #8 is 92.5% larger (H15)
than that of decile #9, and so on.
The largest gap in absolute dollars and in percentages is between
decile #1 and decile #2, a difference of $26.1 billion (G8ÔÇ“G9), or 468.9%
(H8). Deciles #9 and #10 have the second-largest difference between them
in percentage terms (158.2%, per H16). Most deciles are only 45% to 70%
larger than the next-smaller one.
The difference in return (Column B) between deciles #1 and #2 is
1.6% and between deciles #9 and #10 is 3.2%, while the difference between

3
All of the underlying decile data in Ibbotson originate with the University of ChicagoÔÇ™s Center for
Research in Security Prices (CRSP), which also determines the composition of the deciles.

Notes
[1] Derived from SBBI-1999 pages 130, 131.*
[2] SBBI-1999, page 138**
[3] These averages derived from SBBI-1999, pages 200ÔÇ“201.* Beginning of year 1926 yield was not available.
[4] Betas were not available for the 1939ÔÇ“1998 time period.
[5] SBBI-1999, page 140*
[6] CAPM Equation: Rf (Beta Equity Premium) 5.2% (Beta 8.0%). The equity premium is the simple difference of historical arithmetic mean returns for large company stocks and the risk free rate per SBBI 1999 p. 164. The risk
free rate of 5.2% is the 73 year arithmetic mean income return component of 20 year government bonds per SBBI-1999, page 140.*
* Used with permission. 1999 Ibbotson Associates, Inc. All rights reserved. [Certain portions of this work were derived from copyrighted works of Roger G. Ibbotson and Rex Sinque´¬üeld.]
** Used with permission. 1999 Ibbotson Associates, Inc. All rights reserved. [Certain portions of this work were derived from copyrighted works of Roger G. Ibbottson and Rex Sinque´¬üeld.] Source: CRSP University of Chicago. Used
with permission. All rights reserved.
123
F I G U R E 4-1

1926ÔÇ“1998 Arithmetic Mean Returns as a Function of Standard Deviation